Lorentz Transformations are Unable to Describe the Relativistic
Doppler Effect
N. Hamdan
Department of Physics
University of Aleppo –Aleppo –Syria
[email protected]
Abstract
We know paradoxes do not exist in nature and a complete theory does not include them.
We only need to use our intuition to see that there are no logical inconsistencies. In this
context, the relationship between the relativistic Doppler Effect(RDE) and Lorentz
Transformations(LT) exhibits logical inconsistencies. It means that LT misrepresents
reality and describes no physical effects.
In this paper we will explain how to eliminate such logical inconsistencies
(contradictions).
1-Introduction
In his well–known article on "Special Relativity" Einstein succeeded in deriving LT [1],
and then deriving the relativistic Doppler relations based on these transformations.
Therefore RDE is related to the time dilation effect[2]. SRT accounts for various
kinematical effects, like length contraction and time dilation. Several questions arise
when examining this kinematical effect and many contradictions exist. Moreover SRT
and relativistic Doppler relations are incompatible. In this regard we will see that the
asymmetry of kinematical time dilation effect derived by LT makes it difficult to
reconcile LT effects and RDE completely.
In this paper, we depend on the method in [3,4], in which it is shown that the Doppler
calculation procedure as well as interpretation is possibly only with the help of the
Lorentz force law and the relativity principle. This will end the role of the Lorentz
transformation (LT) and of time dilation in RDE.
2-Einstein's Method in Deriving Doppler's Formula :
It is well known that the color of light rays coming out of a moving source towards
the observer tend to be blue shifted (i. e. high frequency), whereas a ray exiting a moving
source in the opposite direction tends to be red shifted (i.e. low frequency).
The diversity of possibilities along with the existence of ether between the source
and the observer leads to four possibilities, explained as follows :
1. If the source is receding/approaching from the rest observer, the frequency that
observer sees is classically
f ′=
f0
1 + (u c)
,
f ′=
f0
(1a,1b)
1 − (u c)
2. If the observer receding/approaching from the rest source, the frequency that
observer sees is classically
u
f ′ = f 0 (1 − )
c
,
u
f ′ = f 0 (1 + )
c
(2a,2b)
By excluding the idea of ether, Einstein has reduced these four possibilities to only two,
namely :
f ′= 1−
u2
c2
1− u
f0
c (3a ) ,
= f0
u
1+ u
1+
c
c
f ′= 1−
u2
c2
1+ u
f0
c (3b)
= f0
u
1− u
1−
c
c
Relativity principle makes it easier to use when it considers these two possibilities as
actually one single possibility, and we get the second possibility through converting the
speed sign in the first.
We can see the difference between the classical Doppler effect applied to light waves and
the RDE. It makes no sense to talk about the velocity of either the source or the observer
relative to the medium as one does in ether. One considers only the relative velocity u
between the source and the observer. As the RDE includes time dilation, i.e. the RDE
includes also the transverse Doppler effect(TDE).
Einstein obtained in his work [1] the following formula:
1 − (u 2 c 2 )
1 + ( u c) cos θ
Where u is the speed of source, θ is the direction of travel.
f ′= f0
(4)
There are two particular cases that lead to simplifications. The first is motion along the
line of sight-the longitudinal relativistic Doppler effect, where.
1-
f ′= 1−
1− u
u 2 f0
c (5a)
=
f
0
2
c 1+ u
1+ u
c
c
Had the source been approaching from the observer then
2-
f ′= 1−
1+ u
u2 f0
c (5b)
=
f
0
2
u
u
c 1−
1−
c
c
The other special case is that of transverse motion across the line of sight. In this case
f
u2
f ′ = f 0 1 − 2 = 0 (5c)
3γ
c
Eqs(5a,5b) have classical analogues in Eqs.(1a,1b).
The frequency is red shifted due to the dilation of the source time, Eq.(5c), and this
effect(TDE) corresponds to the time slowing down on the source moving clock.
In SRT's formalism the key effect for RDE is time dilation, which plays an important part
in modern physics. Therefore Eq.(5c) is considered a unique feature of SRT and is related
to the dilation of time only for the moving source. Einstein believed that the general
formula Eq.(4) which he deduced is an appropriate formula for the two cases, the source
is moving and observer is at rest, or the source is at rest and observer is moving.
As remarked in Einstein's method, the RDE and TRD modes treat only source
receding/approaching from the rest observer. However, an accurate analysis of Eq.(4) by
using LT would reveal that there is an important contradiction between RDE and LT.
According to relativity principle, Eq.(4) could be written for the case of observer in
motion as,
f ′ = f 0 γ (1 −
u
cos θ )
c
If the motion is normal to the line connecting source and observer, we then obtain from
the last equation
f ′ =γ f0
This equation shows a time contraction, instead of a time dilation as in Eq.(5c). We know
that time contraction does not exist in SRT but the symmetry effect of TDR requires a
time contraction.
We will see now that the asymmetry of kinematical time dilation effect derived by LT
makes it difficult to reconcile LT effects and RDE completely.
3- Derivation of RDE and TDE from Lorentz Transformations
Assume two inertial frames S and S ′ , a source with frequency f 0 in the moving frame
S , an observer in the rest frame S ′ , and the source approaching with the relative velocity
u ox from the observer.
The position of the radiation frequency of moving source is described by
x=ct
where ( x , t )
and
Then, we apply LT
( x ′ , t ′)
(6)
x′ = c t ′
,
are spatial and time intervals.
x′ = γ ( x − u t )
, t ′ = γ (t −
u
x)
c2
(7 )
we have
u
t ′ = γ t (1 − )
c
(8)
The frequency is derived as the inverse of time, i.e.:
t=
1
f0
,
t′ =
1
f′
(9)
If we insert Eq.(9) in (8), we 'll find
f ′=
f0
= f0
1 − (u 2 c 2 )
1 − (u c)
1 + (u c)
1 − (u c)
(10)
Had the source been receding from the observer, then by replacing u with
Eq.(17), we have
f ′= f0
1 − (u c)
1 + (u c)
− u in
(11)
Let the reference time in S be t 0 . The reference time in S ′ is defined by Eq.(7), i.e.:
t ′ = γ t0
(12)
Using (9) in (12), we get
f ′=
f0
γ
= f 0 1 − (u 2 c 2 )
(13)
Eq.(10,11 and 13) are the relativistic Doppler shift derived by Einstein [1], but now is
derived from LT directly.
According to relativity principle , we can also consider the frame S to be co-moving with
the source and receding /approaching the observer. Then Eq. (7) could be written as
x = γ ( x ′ + u t ′)
, t = γ (t ′ +
u
x ′)
c2
(14)
Inserting (6) in (14), we obtain
u
t = γ t ′ (1 + )
c
then using (9) in the last equation, we the have
f ′ 1 − (u 2 c 2 )
f0 =
u
(1 + )
c
Or
u
f 0 (1 + )
c
f ′=
2
1 − (u c 2 )
= f0
1+ u
1− u
c
(15)
c
If the observer is receding from the source, then by replacing u with − u in Eq.(15), we
have
f ′= f0
1 − (u c)
1 + (u c)
(16)
If the motion is normal to the line connecting source and observer, we then obtain
f ′=
f0
u2
1 − T2
c
=γ f0
(17)
where in normal movement, the radial component is zero,
u r = 0 , and since
u 2 = u t2 + u r2 = u t2
We obtain also a change in frequency as per Eqs. (17). So the Doppler effect exists even
though there is no component of relative motion along the line of sight.
The reference time of the moving observer becomes long i. e. according to Eqs.(9) and
(17), we have
t ′ = t0 / γ
(18)
The reference time of the moving observer decrease(time contraction), thus the frequency
of the light source that is seen by the moving observer increases as in Eq.(15) and
decreases as in Eq.(16).
Eqs.(15,16 and 17) do not derived by Einstein because it need to time contraction as in
Eq.(18) and the kinematical effect derived by LT is time dilation but not time contraction.
Thus, the Lorentz transformation (7) and (14) have been used for the calculation of the so
called relativistic effect, and we now know that they give the possibility to calculate the
relativistic effect only if LT has time contraction as in Eq.(18).
It means that LT misrepresents reality and reflects no physical effects [3,4].
4- Derivation of RDE and TDE from Lorentz Force
Now assume that a particle q in frame S emits a light wave (photon) that moves in a
direction that makes an angle θ with the positive ox axis. The light is received at the
observer in frame S ′ at an angle θ ′ relative to the ox ′ axis. In [3,4] we have derived the
following relation:
u
v ′ = γv(1 − cos θ )
c
(19a)
But the connection between the two frequencies in frames S and S ′ is given also by
u
v = γv ′(1 + cos θ ′)
(19b)
c
If we consider that the frame S to be co-moving with the source and receding
/approaching observer, Eq. (19a) becomes
u
1 - θ = θ ′ = 0 o i.e. v ′ = v0γ (1 − ) = v0
c
u
c
u
1+
c
1−
(20a)
u
2 - θ = θ ′ = 180 o i.e. v ′ = v0γ (1 + ) = v0
c
u
c
u
1−
c
1+
(20b)
Eqs. (20a,20b) do not have relativistic analogues, but have classical analogues in Eqs.
(2a,2c).
According to relativity principle , we can also consider the frame S ′ to be co-moving
with the observer and receding /approaching source, then Eq. (19b) could be written as
v0 1 −
v′ =
u2
c2
u
1 + cosθ ′
c
(19c)
Hence
u2
u
v0 1 − 2
1−
c
c
1 - θ = θ ′ = 0 o i.e. v ′ =
= v0
u
u
1+
1+
c
c
(21c)
u2
u
v0 1 − 2
1+
c
c
2 - θ = θ ′ = 180 o i.e. v ′ =
= v0
u
u
1−
1−
c
c
(21d)
Eqs. (21c,21d) are identical to Eqs. (5a,5b) in SRT, and the classical analogues are (1a,
1b).
If the velocity of the observer/ source is perpendicular to the line of sight, then we have
from Eq. (19a),
v0
θ = θ ′ = 90 o i.e. v ′ =
= γv0
(22a)
u2
1− 2
c
And from Eq. (19c), we have
θ = θ ′ = 90 o i.e. v ′ = v0 1 −
u 2 v0
=
γ
c2
(22b)
In [3] we have shown, that had it not been for the existence of formula (22a) and
(22b) together, there would be no equality between the two formulas (20a) and (21c), and
the two formulas (20b) and (21d). This means that formula (22a) exists since this formula
is not the outcome of SRT due to time contraction. This would mean that LT is unable to
describe a well – known physical reality, namely Doppler effect.
We turn to Eqs.(19a,19c) i.e.
u2
v0 1 − 2
u
c
′
′
v = γv0 (1 − cos θ ) , and v =
u
c
1 + cos θ ′
c
(23)
The two angles in Eqs.(23) differ when we include the effect of aberration. If we let
θ denote the angle with respect to the source’s frame and θ ′ denote the angle with
respect to the observer’s frame, then we have
u2
v0 1 − 2
u
c
= γv0 (1 − cos θ )
u
c
1 + cos θ ′
c
Or
u2
1− 2
u
c
1 + cos θ ′ =
u
c
1 − cos θ
c
cos θ −
i. e.
cos θ ′ =
u
c
u
1 − cos θ
c
( 24)
Eq.(24) describe the aberration of light. Star aberration arises because the observe
moves with the orbital speed of the Earth. If , as SRT asserts, the movement of the light
source is equivalent to the movement of the observer, star aberration has to arise as in the
case when the source moves. However, the observations of binary stars prove that there is
no aberration when the stars move.
Conclusion:
Due to the Eqs.(22a) and (22b) together, the RDE formula for a moving observer can also
be written in the form used for a moving source.
Certainly formula (22a), which does not have an equivalent in the SRT, is very
significant for the formula (22b) for the equality of the longitudinal relativistic Doppler
effect. That means the longitudinal relativistic Doppler effect for a moving observer can
be also written in the form used for a moving source.
The kinematical effect derived by LT is time dilation but not time contraction. Thus, the
asymmetry of the kinematical time dilation effect derived by LT makes it difficult to
reconcile LT effects and RDE completely. These contradictions cannot be removed
without excluding LT and deriving RDE and TRD formula without LT [3,4].
Star aberration and the TDR arise as the result of using the Lorentz force and relativity
principle. Both effects do not prove that time dilation takes place in moving systems.
Both these phenomena contradict SRT and prove it is false.
References
[1] A. Einstein, “On the Electrodynamics of Moving Bodies”, Ann. Phys. 17, 891
(1905).
[2] R. Ferber, “A Missing Link: What is behind de Broglie’s periodic phenomenon?”,
Foundations of physics Letters 9 (6), 575–586 (1996).
[3] N. Hamdan, “Derivation of the Relativistic Doppler Effect from the Lorentz
Force", Apeiron, Vol 12, No. 1, Jan.(2005).
[4] N. Hamdan, “On the Interpretation of the Doppler Effect in the Special
Relativity(SRT)", Galilean Electrodynamics, March/April , Vol. 17, No. 2. pp.29-34,
(2006).
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